Abstract. We prove that the countable intersection of C 1 -diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in R d , improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt's (α, β)-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs.

Given an integer matrix M∈GLn(ℝ) and a point y∈ℝn/ℤn, consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y∈ℚn/ℤn, the set $ \tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M∈GLn(ℝ)∩Mn×n(ℤ) and y∈ℝn/ℤn, and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m×n matrices. Furthermore, we show that sets of the form $ \tilde E(M,y)$ and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝn. As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint, arXiv:0912.2445] on badly approximable systems of affine forms.

We explore and refine techniques for estimating the Hausdorff dimension of exceptional sets and their diffeomorphic images used in [3], which were motivated by the work of W. Schmidt [25], D. Kleinbock, E. Lindenstrauss and B. Weiss [17] and C. McMullen [22]. Specifically, we use a variant of Schmidt's game to deduce the strong C 1 incompressibility of the set of badly approximable systems of linear forms as well as of the set of vectors which are badly approximable with respect to a fixed system of linear forms. This generalizes results in [2], [10], and [3].

Abstract. Given b > 1 and y ∈ R/Z, we consider the set of x ∈ R such that y is not a limit point of the sequence {b n x mod 1 : n ∈ N}. Such sets are known to have full Hausdorff dimension, and in many cases have been shown to have a stronger property of being winning in the sense of Schmidt. In this paper, by utilizing Schmidt games, we prove that these sets and their bi-Lipschitz images must intersect with 'sufficiently regular' fractals K ⊂ R (that is, supporting measures µ satisfying certain decay conditions). Furthermore, the intersection has full dimension in K if µ satisfies a power law (this holds for example if K is the middle third Cantor set). Thus it follows that the set of numbers in the middle third Cantor set which are normal to no base has dimension log 2/ log 3. IntroductionLet b ≥ 2 be an integer. A real number x is said to be normal to base b if, for every n ∈ N, every block of n digits from {0, 1, . . . , b − 1} occurs in the base-b expansion of x with asymptotic frequency 1/b n . Equivalently, let f b be the self-map of T def = R/Z given by x → bx, and denote by π : x → x mod 1 the natural projection R → T. Then x is normal to base b iff for any interval I ⊂ T with b-ary rational endpoints one haswhere λ stands for Lebesgue measure.É. Borel established that λ-almost all numbers are normal to every integer base; clearly this is also a consequence of Birkhoff's Ergodic Theorem and the ergodicity of (T, λ, f b ).Note that it is easy to exhibit many non-normal numbers in a given base b. For example, denote by E b the set of real numbers with a uniform upper bound on the number of consecutive zeroes in their base-b expansion. Clearly those are not normal, and it is not hard to show that the Hausdorff dimension of E b is equal to 1. Furthermore, it was shown by W. Schmidt [26] that for any b and any 0 < α < 1/2, the set E b is an α-winning set of a game which later became known as Schmidt's game. This property implies full Hausdorff dimension but is considerably stronger; for example, an intersection of countably many α-winning sets is also α-winning (we describe the definition and features of Schmidt's game in §3). Thus it follows that the set of real numbers x such that for each b ∈ Z ≥2 their base-b expansion does not contain more than Date: January 2010.

Abstract. The set of badly approximable m × n matrices is known to have Hausdorff dimension mn. Each such matrix comes with its own approximation constant c, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed c. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions. The lower bounds are established via the technique of Schmidt games, while for the upper bound we use homogeneous dynamics methods, namely exponential mixing of flows on the space of lattices.